Models and Math

[cesium62]

The models are tuned over many runs to make them consistent with both science and history.
[...]
Much harder, in point of fact, than fitting a curve to some data.

Bruce, I nearly totally agree with you. But keep in mind that "tuning parameters in a model to make it consistent with history" and "fitting a curve to some data" is in every aspect the same thing...

That is reverse physics. Instead of testing the model (which should be a mathematical simplified describtion of some aspects of the nature) and rejecting a model if not fitting to observations you may tune the parameters every year again. There is no real value or deeper understanding of the world gained by doing the latter - that is only usefull for politics or convincing poeple for a very short time (and maybe that ugly standard model, which you could also fit to an elephant...).

The model informs the shape of the curve that you use.  F = m1*m2/d^2 is a model.  You still want to go out and measure m1, m2, and d for a particular application of the model.

That formula is the direct result of a theory. It is not a model. A moel would be to approximate the force and make it constant when distance is very small, to avoid too large errors near the d=0 singularity. That is THE model that can be used in computer simulations. The actual formula cannot be used for many calculations due to its singularity.
Guys please is this stuff for this thread?

Math is a model.  You want to show me where m1 and m2 and d exist in isolation?  When the distance is not very small, the formula is the formula that you would use for many calculations.  Is the distance usually not very small?  Are most pairs of objects in the universe nearly on top of each other?

[cesium62]

The models are tuned over many runs to make them consistent with both science and history.
[...]
Much harder, in point of fact, than fitting a curve to some data.

Bruce, I nearly totally agree with you. But keep in mind that "tuning parameters in a model to make it consistent with history" and "fitting a curve to some data" is in every aspect the same thing...

That is reverse physics. Instead of testing the model (which should be a mathematical simplified describtion of some aspects of the nature) and rejecting a model if not fitting to observations you may tune the parameters every year again. There is no real value or deeper understanding of the world gained by doing the latter - that is only usefull for politics or convincing poeple for a very short time (and maybe that ugly standard model, which you could also fit to an elephant...).

Fitting curves to a set of data points and tuning parameters to a model are nowhere near the same thing. The amount of freedom you have in choosing curves is large.  For a simple model, the model can be validated in simple circumstances.  For a complicated model, pieces of the model can be validated in simple circumstances.  When you pick a simple model to use, it is like choosing the type of curve that you fit to your data.


Sure some problems don't have good models.  Some problems require higher resolution than we can obtain to provide good accuracy.  And yet, we can still usually tell whether a hurricane will travel north or south.

You don't have a version of reality sitting inside your brain.  You have a model of reality.  You use models every day.  You use them to figure out what people are saying; you use them to figure out what people might be thinking.  So don't go dissing all models too heavily.

[cesium62]

I'm sorry but saying the models "only work as long as the future looks like the past" is a total misconception. The models are physical models based on physics worked out theoretically and underpinned, in some cases 'tuned by', obervational work. They do not depend on the past as a guide, they use physics. [...]

This is a dramatic overstatement of the truth. The models have numerous parameters, most of which are only approximately known. Look at the error bars on total climate sensitivity -- they're enormous. The models are tuned over many runs to make them consistent with both science and history.

I'm not suggesting that the models are junk or that they should be thrown out with the bathwater. The GCMs are quite good, and have accurately predicted a number of important features of climate change. But the idea that they are boxes of pure applied physics is not remotely true. Like all models, they are approximations of reality. And where approximations are made, uncertainty creeps in. No new physics or magic is required. Simply not properly accounting for changes in parameters or relationships is enough to send one's model wildly astray. In a rapidly changing system, it is very difficult to remain on track as you model into the future. Much harder, in point of fact, than fitting a curve to some data.

Over the long run, you can refine a model and gain more accurate resolution of the input data to the model.  You can prove (find strong evidence for) more and more pieces of a complicated model.

Over time, fitting curves to data is still fitting curves to data.

[ChrisReynolds]

The models are tuned over many runs to make them consistent with both science and history.
[...]
Much harder, in point of fact, than fitting a curve to some data.

Bruce, I nearly totally agree with you. But keep in mind that "tuning parameters in a model to make it consistent with history" and "fitting a curve to some data" is in every aspect the same thing...

That is reverse physics. Instead of testing the model (which should be a mathematical simplified describtion of some aspects of the nature) and rejecting a model if not fitting to observations you may tune the parameters every year again. There is no real value or deeper understanding of the world gained by doing the latter - that is only usefull for politics or convincing poeple for a very short time (and maybe that ugly standard model, which you could also fit to an elephant...).

Fitting curves to a set of data points and tuning parameters to a model are nowhere near the same thing. The amount of freedom you have in choosing curves is large.  For a simple model, the model can be validated in simple circumstances.  For a complicated model, pieces of the model can be validated in simple circumstances.  When you pick a simple model to use, it is like choosing the type of curve that you fit to your data.....

Exactly why I 'groaned', at Bruce. I was so exasperated I couldn't formulate a polite reply.

Thanks for picking up where my temper left off. 

[SATire]

cesium62, it is a bit hard to understand your questions or what exactly I could explain in more detail. But I am going to commit myself to explain the math-things since I proposed such "simple-math" thread to keep that outside of the melting season.

As typical in math: I would like start with the definitions. After that we can start proofing/disproofing arguments (math arguments - not the conversation thing).

"Math is a model. " No - math is a logical construction and not made to describe the world in any way. Physics uses math as a language and a tool to compute things to describe (=model) the most simple aspects of the nature. (Different from your experience in school mathematicans do not compute but proof/disproof arguments).

"Fitting curves to a set of data points and tuning parameters to a model are nowhere near the same thing. The amount of freedom you have in choosing curves is large.  For a simple model, the model can be validated in simple circumstances.  For a complicated model, pieces of the model can be validated in simple circumstances.  When you pick a simple model to use, it is like choosing the type of curve that you fit to your data."

Fitting curves and tuning parameters of a model is the same thing. The word "fitting" means, to tune one or several parameters until the function/model fits to the data. A function may be used as a model. But a model may be complicate and consist of many functions or be the result of a system of coupled non-linear differential equations. Not much of a difference from mathematical point of view - only different effort to solve that practically.

And if you keep on groaning you are right - math is simple and boring compared to all the things happening in nature...

[SATire]

Math is a model.  You want to show me where m1 and m2 and d exist in isolation?  When the distance is not very small, the formula is the formula that you would use for many calculations.  Is the distance usually not very small?  Are most pairs of objects in the universe nearly on top of each other?

To get physical with "F = m1*m2/d^2": d is here the distance between the center of mass of one thing to the center of mass of the other thing. As long as the one thing is not inside/at the same place like the other thing you are safe.

[Jim Hunt]

For those who might wish to explore alternative sea ice models:

http://forum.arctic-sea-ice.net/index.php/topic,624

For those who might wish to experiment with current sea ice models:

http://forum.arctic-sea-ice.net/index.php/topic,108

[Xyrus]

...
Fitting curves and tuning parameters of a model is the same thing. The word "fitting" means, to tune one or several parameters until the function/model fits to the data. A function may be used as a model. But a model may be complicate and consist of many functions or be the result of a system of coupled non-linear differential equations. Not much of a difference from mathematical point of view - only different effort to solve that practically.

And if you keep on groaning you are right - math is simple and boring compared to all the things happening in nature...

"Tuning" is adjusting the parameters based on observation/data/what have you to have a more accurate model. Tuning changes the values, but it doesn't change the physics, and more importantly you CAN'T change the physics without some damn good justification for it.

There is no physical model for curve fitting. You can make up any equation you want, use whatever parameters you want, and come up with something that you think is the best fit. However, there is no rhyme or reason. There is no predictive power. A fitted curve is just that, a fitted curve. With a validated context it might have some predictive power, but otherwise it's just a curve through some data.

[SATire]

Oh Xyrus,
I am sure you know that a physicist would only fit a function to some data if there is a reason - that means that the function is his model. And the reason to fit is to check, if he has to reject that function (model) or if it could still be left in the game.

If there would be no model, he could just draw some lines between the data points and call that the function (that is actual really a function). Fitting without a model does not really make any sense, since there is no model to check. For such games we have splines and such.

May problem with fitting and data tuning is, that you use the available data to fit/tune the free parameters. So what is left to test your model? No data anymore - only some hope for future data...

OK - you probably know, that a lot of statistics can be used to test the fit/model still. But try to explain that your neighbor...

[cesium62]

cesium62, it is a bit hard to understand your questions or what exactly I could explain in more detail. But I am going to commit myself to explain the math-things since I proposed such "simple-math" thread to keep that outside of the melting season.

As typical in math: I would like start with the definitions. After that we can start proofing/disproofing arguments (math arguments - not the conversation thing).

"Math is a model. " No - math is a logical construction and not made to describe the world in any way. Physics uses math as a language and a tool to compute things to describe (=model) the most simple aspects of the nature. (Different from your experience in school mathematicans do not compute but proof/disproof arguments).

"Fitting curves to a set of data points and tuning parameters to a model are nowhere near the same thing. The amount of freedom you have in choosing curves is large.  For a simple model, the model can be validated in simple circumstances.  For a complicated model, pieces of the model can be validated in simple circumstances.  When you pick a simple model to use, it is like choosing the type of curve that you fit to your data."

Fitting curves and tuning parameters of a model is the same thing. The word "fitting" means, to tune one or several parameters until the function/model fits to the data. A function may be used as a model. But a model may be complicate and consist of many functions or be the result of a system of coupled non-linear differential equations. Not much of a difference from mathematical point of view - only different effort to solve that practically.

And if you keep on groaning you are right - math is simple and boring compared to all the things happening in nature...

My questions are rhetorical.

In math, we explicitly use the word model.   A model does not need to be a model of physical reality.  In computer programming, we also strive for well defined mathematical models so that we can reason about our program behavior.  Don't assume you have any idea as to my experience in school.  And don't patronize me.  That entire paragraph smells of troll.

In your fifth paragrpaph, thanks for not reading nor responding to what I wrote; thanks for repeating yourself without clarifying or adding to the conversation.

And try not to mix Chris and I up.  He is Cr.  I am Cs.

[cesium62]

For those who might wish to explore alternative sea ice models:

http://forum.arctic-sea-ice.net/index.php/topic,624

For those who might wish to experiment with current sea ice models:

http://forum.arctic-sea-ice.net/index.php/topic,108

Ah, there always has to be one adult in the room.  ;-)  Trying to keep this thread civil and high brow I see...

More seriously...  thanks for the links.  I'll work on exploring those in more detail...

[cesium62]

Math is a model? (tell that to a pure mathematician!). You meant the other way around probably.

As a sometime pure mathematician, I probably meant what I wrote.

[SATire]

Hi everybody.

The comment from Xyrus reminded me of the fact, that "curve fitting" is done for 2 different purposes:
1) For testing a function as a model, if it could fit to the data with some confidence or not.
2) For drawing a scientific looking line through some points e.g. to suggest something. I do not like the latter but I have to accept, that this exists. So it was my mistake to ignore No.2 persistently.
So if I stated "parameter tuning" and "fitting" is the same, that was stated only for No 1 case. That statement was completely wrong if related to case No 2.

I would kindly like to ask the "curve-fitters without model to test" to clearly specify that. E.g. I was told in scientific papers to clearly mark such curves as "guide to the eye" - since that is the main purpose of such curve fitting: To guide the reader to follow some suggestion - and some kind of marketing, to make the graph with all the noisy strange data scattering looking a bit more "scientific".

In my view "curve fitting without model" can not be used for predictions and does not help in understanding. This is usefull either for interpolation between data points (does not make sense e.g. for min. ice extent and such), for cosmetics (nice looking) or for intended suggestion (guide to the eyes) - which should be clearly specified to avoid to be accused of intended manipulation of the reader.
Furthermore I would suggest the curve fitters to quit searching for nice looking functions but to use a device like in the attached photo: That things allows to draw suggestive lines to any data with maximum precision.

[Rick Aster]

I would kindly like to ask the "curve-fitters without model to test" to clearly specify that. E.g. I was told in scientific papers to clearly mark such curves as "guide to the eye" - since that is the main purpose of such curve fitting: To guide the reader to follow some suggestion - and some kind of marketing, to make the graph with all the noisy strange data scattering looking a bit more "scientific".

As this is a forum in which participants have a wide range of skill at science, I would hope for some indulgence for writers who fit a curve, or extrapolate in some manner, just to imply a hypothesis, or to look for one. If nothing else, a fitted curve implies that the two variables are probably not independent of each other. To my mind, it is perfectly acceptable to note that two variables seem to be related without first having to know the nature of the dependency.

[Xyrus]

Oh Xyrus,
I am sure you know that a physicist would only fit a function to some data if there is a reason - that means that the function is his model. And the reason to fit is to check, if he has to reject that function (model) or if it could still be left in the game.

If there would be no model, he could just draw some lines between the data points and call that the function (that is actual really a function). Fitting without a model does not really make any sense, since there is no model to check. For such games we have splines and such.

May problem with fitting and data tuning is, that you use the available data to fit/tune the free parameters. So what is left to test your model? No data anymore - only some hope for future data...

OK - you probably know, that a lot of statistics can be used to test the fit/model still. But try to explain that your neighbor...

We're clearly not on the same page here.

A physics model exists and works REGARDLESS OF THE DATA. Hence, the equation for gravitational force works just as well for the moon as it does the Earth. The only thing that is different is the value of the parameters. A physical model describes a real world physical relationship. Tuning values of parameters to get a better match to observations does not alter this relationship. The model is still valid and still makes useful predictions, it's just that now it can do an even better job because you gave it better data.

This is vastly different from curve fitting. While curve fitting can, with enough work and validation, demonstrate that a possible physical relationship can exist, it does not mean that there is one or that it is even correct.

[SATire]

We're clearly not on the same page here.

A physics model exists and works REGARDLESS OF THE DATA. Hence, the equation for gravitational force works just as well for the moon as it does the Earth. The only thing that is different is the value of the parameters. ...

Xyrus, I agree that we disagree...

What you describe sounds like engineering to me but not like natural sciences. In engineering you assume that a physical model exists and is proofen and works - so you have no need to test that. So engineers only need to get the parameters like mass of the planet - which they may obtain by fitting the equation for gravitional force to some data like position of moon vs. time. So they may "tune" the parameter "mass" by fitting an equation to some data and then they are fine. My no means they can test, if the equation for the gravitation is "true" by this procedure.

In natural science instead a sentence like "A physics model exists and works REGARDLESS OF THE DATA" is most wrong. Instead I would formulate: A physical model may work until it is prooven wrong - e.g. if the model can not fit to the data within e.g. 3 sigma. If a physical model is not prooven wrong, it can be used. But it can not be prooven right. Never. The truth is for philosophy. In natural science we can "only" say this model works in this regime - but "model" "works" and "regime" are quantified very precisely and can be measured anytime and everywhere by everybody. They test "gravitational force" every day to make a better model including dark matter and search for gravitational wave data...

And also the meaning "parameter" is different in natural science: E.g. in physics we obtain the parameter "mass" of a Higgs particle by fitting the standard model to data at CERN - and if we obtain a peak 5 standard deviations away from both statistical errors and the systematic errors obtained by monte carlo simulation of the measurement equipment, than we say it is save to assume the model "there is a Higgs particle with this mass" to be valid - so far, until someone comes up with a more precise measurement disprooving that in future. But a model in contradiction to observation (data) is an ex-model immediately (but may be used for simplicity in special conditions - like e.g. Newtons formulars in the case of small velocities).

After your statement I have the feeling, that the simulation in climate science may be closer to my above mentioned No. 2 (curve fitters without model) than to No1. (testing a model). I hope I am wrong and that you may take your time to explain your view.

[ChrisReynolds]

SATire,

In engineering we do generally assume a physical model exists is proven and works. We generally have a toolbox of applied physics, and when something varies from the physics we apply, we often assume there is an extraneous factor that needs to be identified and removed, rather than that the physics are wrong. However not every problem has a solution*, and the more complex one gets the more this principle must be kept in the back of one's mind. *within budgets of time and money.

I'm more interested in things from the point of view of an engineer. I get bored with the scientific philosophy that occupies scientists. I'm interested in whether models are a useful tool, that is how I view physics and math - can it help me get the job done?

So I ask questions of a model...

Does the model produce output that is a 'good fit' (often involves subjective judgment) to real observations? Here producing output that is statistically indistinguishable from observations, not necessarily identical, is the best test.

If the model has been tuned to better fit real observations is this done only within reasonable limits of what the evidence suggests the tuned parameters actual values may really be? In complex scientific situations - the real world - you are forced to make such guesses, they should be defensible and realistic.

Does the model make predictions? Not just predictions of what comes next, but does the model predict, for example, physical quantities that can be verified.

Crucially, is the model 'fit for purpose', does it prove to be a useful tool? A critical feature here in engineering is a model that, when observations don't fit the model, tells me to examine the set up I'm using because there is a problem in it. In engineering the most common example of this is the humble thermocouple. In climate science it's when models say the tropical troposphere should be warming, Spencer and Christy say it isn't, on deeper examination of the satellites the models are shown right, Spencer and Christy wrong.

Climate models fit the bill in all the above questions. Curve fitting doesn't.

Curve fitting is what you do when you don't know anything about the processes involved. It may be a useful bridge to figuring out the processes, but it is not the end of the process. Sadly I consider that for many in the amateur sea ice community the curve fitting to PIOMAS volume has become the 'be all and end all'.

[SATire]

Chris, thank you for this very constructive comment.

Beyond discussion about disciplines (philosophy, physics, engineering, ...) we should keep an eye on the goal. However, I think climate science is an interdisciplinary science. Thus, it is essential to know the different meaning of words in different disciplines during discussions. That knowledge about the other disciplines is essential for understanding each other - if something is boring to you please try to hide that feeling because that may be offensive for your partner in discussion and could result in not helpful emotions. Most of the troubles in forums can be related to such emotions. Thus I want to try to describe my points of view in technical language:

About curve fitting.

In physics curve fitting can be used for both: Getting parameters by fitting a model to data and also checking, if that model could be valid and must not be rejected. But you must be very careful, that you can do all the necessary steps:
1) you know the error bars in your data before you start any fitting. You must know precisely the statistical error - you have to quantify the noise. You must give values for your systematic errors resulting from the way you measure and from the measurement apparatus. You can specify the data with both errors - e.g. like  y(5.6)= 3.7 +- 0.4 statistical error +0.3 -0.5 systematical error.
2) you have a theory to test which can describe your problem to be solved as a function y=f(x) you want to test.
3) you do the fit and know about the covariance matrix or covariance ellipse to get the uncertaincy of the parameter you want to fit to the data
4) you know that chi^2 should be e.g. number of data - degrees of freedom and you can compute the significance of that fit to jugde, if you have to reject your model or if your measurements are still not precise enough to do that.

If you are not able to survive the first step in any later discussion you are out and blamed for ever... Like the "neutrinos traveling faster than light"-guy. He got retired.

In engineering: As you explained you may fit a model to data to get a parameter. If you know about the errors in your data you also get the error bars for your parameter and you may work with that. This is usefull but you can not make any predictions by doing so. Next year poeple may do the same work and the value of the parameter might be a bit different. But that is ok - since you only wanted to get that parameter and it was the best value you could obtain at that date. So please work with that value for which ever goal you needed that parameter.

To draw a line: Please do not try cuve fitting but use the curve stencil to draw that line. That is way more honest and way more precise, after you learned to move that curve stencil dynamically over your data points to draw very precise curves through intervals of your data. This is not a bad thing if used for drawing a nice-looking line to your data. But please do not blame yourself by making any predictions or theories from that line. E.g. a correlation test may deliver more conclusions than that line. But useful that line still is for some purposes.


To get back to our problem in climate science: Here we try to use physical models to describe a complex thing. However - physics did only care about the simple things. So now we do have plenty models of boring simple things. But we do want to predict the future starting with a lot of vague known parameters and a lot of physical models precisely in a space and time grid. Since you can not put all quantum mechanics into that problem and compute the systems in any time smaller than the number of atoms in the universe you have to neglect most of the physics. But which ones? The less important ones. How to know about which interdependencies to neglect savely? This is the point where my critics about fitting in complex models arrises:

In this kind of engineering we need some curve fitting to test, if we have choosen the most important interdependencies in the physical models we choose to simulate the climate. But if we used the available data to tune/fit some unknown parameters allready - what do we have left to test our choice? By putting the methodes of engineering and physics together we are now left to justify our result on the most complex physics side - that is very difficult and a very weak point. For me that is the main problem in explaining climate science and I would be happy to learn about concepts to solve that. Those concepts could be available from climate scientist like Xyrus and I hope to learn something here about that.

Kind regards,
Jens

[Rick Aster]

Curve fitting is what you do when you don't know anything about the processes involved.

I think this assertion is only partly right, and is the gist of the problem here. You can have a pretty good guess at the processes involved — thinking of the change in the moon’s mass as a physical example, where water, dust, and impacts are known and reasonably well-understood events — yet still get your best measure of a process of change by curve fitting. It is not that “you don’t know anything” but that it is too expensive to collect enough of the intermediate data to do a statistically meaningful systematic analysis.

If I seem strident, consider my predicament as an economist. For some of the most interesting things that economists look at, curve fitting is virtually all we can do. We know what a lot of the underlying processes are, or at least we think we do, but there isn’t much we can directly measure. When we fit curves we might imagine we are fitting a model but most of the models that economists tout can’t be meaningfully tested.

[cesium62]

Math is a model? (tell that to a pure mathematician!). You meant the other way around probably.

As a sometime pure mathematician, I probably meant what I wrote.

Ok then probably you know what I mean.

Tell me son, is the euclidean geometry property alpha + beta + gamma= π, where, alpha beta gamma are the angles of a triangle, a model per se? No. It is a mathematical certainty
However, if you use euclidean geometry (assume Earth is flat) to obtain the bearing angle to follow in order to reach London from Normandy, by knowing the bearing to Bristol (thx to a navigation aid) and knowing the flying distance Bristol-London, that is a model. And a useful one because errors of considering Earth flat are small for these distances. However errors grow cuadratically with distance and the model no longer useful for oceanic distances.

(Just kidding calling u son, I am sarcastic of my own tone, Im patronizing anyways Im too old. Sorry for that!)

I'm probably flashing back on something related to http://en.wikipedia.org/wiki/Model_theory ...

[Michael Hauber]

All physics is based on curve fitting.  We determine that gravity follows a formula such as F=GMM/r2, where G is the gravitational constant.  How do we determine this constant?  Do an experiment, get some data, fit a curve to the data and obtain the parameter from this curve.

Even F = MA is really F = kMA, where k is some type of constant.  Except we define the units of Force explicitly so that K = 1 - One newton is defined as the force required to accellerate 1 kg by 1 m/s2.  If instead 1 Newton was defined as the Force exerted by gravity for two 1kg objects at 1 metre seperation we would define gravitational force as F = MM/r2 without a constant, and have to do a curve fit exercise to find the  constant for F = kMA.

Of course in this case we have an advantage over blind curve fitting of supposedly knowing that the form of the equation is F=kMA, and that we can fit only a linear curve, and not a polynomial or exponential.  Except for the fact that Newton's laws of motion are only accurate over a particular range and if you extrapolate to far into unknown conditions you find the formula is no longer accurate and you have to use relativity.

So F=MA.  Or extrapolating a polynomial function for minimum volume in the Arctic.  Its all based on curve fitting, and if you extrapolate too far you get into trouble.  Its just a matter of how far is too far.

[SATire]

from that model theory link above: "We call a theory a set of sentences in a formal language, and model of a theory a structure (e.g. an interpretation) that satisfies the sentences of that theory".   
As usual in mathematics words obviously have completely different meaning than in physics, engineering or normal life. The math poeple may define everything they like and nobody can stop them - they even proof that god exists so there is no need anymore to believe: http://arxiv.org/abs/1308.4526 

But the math poeple can not prevent the physics poeple from grabbing every little thing and making some use of it for modelling the world 

So for math poeple math is a set of logical constructions. For most others math is a set of nice (and also some less nice) tools.

"math is a model" probably is a rhetoric joke in context with the rhetoric questions - probably used by caesium137 with the goal to seed some discussion. That worked nicely. 

[ChrisReynolds]

SATire,

Thanks for taking the time to explain so carefully. Please don't take offense at my description of scientific philosophy as boring. There is an English phrase, 'horses for courses', the French equivalent being 'À chacun son goût', just because I consider something boring doesn't mean I am dismissing either it, or the people who don't find it boring.

I'm not dismissing curve fitting, indeed I did say: "It may be a useful bridge to figuring out the processes, but it is not the end of the process."  And I have indulged in a set of equations from three fits to scatter plots to try to throw light on the future. However I think we were on the same page with regards curve fitting anyway.

But if we used the available data to tune/fit some unknown parameters allready - what do we have left to test our choice?

In PIOMAS the submarine transect (Data Release Area) data, and the ICESat data were split into two groups in space and time domains. One set was used to tune the model, the other for testing. This might be considered dubious because of similar behaviour in the two datasets (is the term autocorrelation?). Anyway a further test was available recently when after years from the last ICESat training, the Cryosat 2 data was compared with PIOMAS and the fit found to be reasonable.

This method of seperating the tuning data and the testing data is, as far as I understand, common practice for modellers. Hopefully Xyrus can comment on that.

But the math poeple can not prevent the physics poeple from grabbing every little thing and making some use of it for modelling the world.

And the physics and the maths people cannot stop engineers from putting their beautiful work into nuts and bolts products to make money. 

SeaIceSailor,

It's been years since I did my degree, I have only vague memories of examples where physicists started out with curve fit equations and worked towards the correct underlying physics - but I do have recollections. Perhaps I was a bit hard, but to add to your comment about PIOMAS curve fitting...

Curve fitting to PIOMAS volume data is making a fit to the output of a model, finding that it results in a zero crossing at 2016 +/-3 years, while ignoring that when the model itself is used to forecast (with random weather plus a warming trend from IPCC scenarios) it shows that autumn growth of ice severely reduces overall volume loss, and significantly delays the 'zero crossing'.

Cesium has taken your equations over to the slow transition thread, I've questions over there.

[Xyrus]

All physics is based on curve fitting.  We determine that gravity follows a formula such as F=GMM/r2, where G is the gravitational constant.  How do we determine this constant?  Do an experiment, get some data, fit a curve to the data and obtain the parameter from this curve.

All physics is based on reality. The arbitrary nomenclature we refer to as "math" is just how we encode it in a form we can understand and use. Regardless of how we encode it the actual physical phenomena remains the same. That's the difference between "curve fitting" and "physical model".

Even F = MA is really F = kMA, where k is some type of constant.  Except we define the units of Force explicitly so that K = 1 - One newton is defined as the force required to accellerate 1 kg by 1 m/s2.  If instead 1 Newton was defined as the Force exerted by gravity for two 1kg objects at 1 metre seperation we would define gravitational force as F = MM/r2 without a constant, and have to do a curve fit exercise to find the  constant for F = kMA.

Of course in this case we have an advantage over blind curve fitting of supposedly knowing that the form of the equation is F=kMA, and that we can fit only a linear curve, and not a polynomial or exponential.  Except for the fact that Newton's laws of motion are only accurate over a particular range and if you extrapolate to far into unknown conditions you find the formula is no longer accurate and you have to use relativity.

The form of the equation is irrelevant. It's a convenient way for expressing the phenomena. That's all.

Relativity and Newtonian mechanics are completely consistent with each other. For relative reference frames at low velocities, the more complicated terms from the relativistic equations become essentially zero (resulting in Newton's equations). They are not unrelated.

So F=MA.  Or extrapolating a polynomial function for minimum volume in the Arctic. Its all based on curve fitting, and if you extrapolate too far you get into trouble.  Its just a matter of how far is too far.

This is simply not true. Something like F=MA is describes a real physical manifestation that has been observed and verified over centuries. You don't "extrapolate" anything. It's a validated predictive model. But there are known limitations, at which point you need to switch to higher order model (relativity).

Fitting a polynomial curve to arctic sea ice volume has no physical basis. The polynomial may be a "best fit" but there's no validation and no theoretical basis for WHY it is a best fit. It is not a predictive model. It has absolutely no valid predictive capability. Any prediction you make based on the curve is scientifically worthless since there's absolutely nothing to back it up with. Yes, it happens to fit. But why? What's the physical mechanisms driving it? If you don't understand the mechanism for WHY something is happening, curve fitting is little more than eye candy. It might guide you into avenues of investigation but by itself it doesn't explain anything.

That's the key difference between scientific results and the graphturbation you see on WUWT and the other nutter sites. 

[Michael Hauber]

So what is the physical basis for F=MA?

There is no way to show a physical basis for this except to go collect some data and fit a curve to it.

[viddaloo]

So F=MA.  Or extrapolating a polynomial function for minimum volume in the Arctic.  Its all based on curve fitting, and if you extrapolate too far you get into trouble.  Its just a matter of how far is too far.

If you take this PIOMAS graph, it tells me that ice volume loss by the end of August for the next 5 years — 2015 to 2019 — will likely be a lot closer to the green trend line than 2014 was. As 2014 at 69% is a low extreme, all 5 years will probably be above 70%. One or two of the years will probably be above the green line, and then possibly even above 85%.

So I think I'm using trend lines sensibly when I use them to tell signal from the random noise («weather»). Remembering Tyson from Cosmos II: «Keep your eye on the man, not the dog.»

[SATire]

Chris,

I did not feel offended by any of your describtions - that was a more general comment also to remind me to try be more careful - so: np.

But if we used the available data to tune/fit some unknown parameters allready - what do we have left to test our choice?

In PIOMAS the submarine transect (Data Release Area) data, and the ICESat data were split into two groups in space and time domains. One set was used to tune the model, the other for testing. This might be considered dubious because of similar behaviour in the two datasets (is the term autocorrelation?). Anyway a further test was available recently when after years from the last ICESat training, the Cryosat 2 data was compared with PIOMAS and the fit found to be reasonable.

This method of seperating the tuning data and the testing data is, as far as I understand, common practice for modellers. Hopefully Xyrus can comment on that.

In the PIOMAS case as far as I understood that thing it is performed like in normal physics curve fitting: They can both test the quality of their choice of models with experimental data and also obtain some parameters with error bars - and you may use those parameters (e.g. regional thickness) for your nice engineering. But you know that they do not make any predictions about next years thickness. There is no physics in PIOMAS backing up any predictions about medium time scale future change of parameters like sea ice thickness as far as I know. So please be careful if you would like to try that.

While you may handle a boundary value problem it is very difficult to do the initial value problem in this very dynamic/non-linear/complex case
 
My problems with simulations are more related to the class of models simulating future climate change. It is thus very complicated to explain them and to obtain some confidence about their quality.

[SATire]

So what is the physical basis for F=MA?

There is no way to show a physical basis for this except to go collect some data and fit a curve to it.

Sorry I double post.
I'd go farther and ask: so what can be a physical basis apart from F=MA?. There are not many.

Seaicesailor - in the main point Michael is right: The experimental observation is the sole thing to judge any model in physics - this is a natural science and not arts like math.

If you look back about mechanics you know, that very often you obtain F=m v in observations. Two man dragging a stone to build the temple move twice as fast as one. The reason is, friction is the most important thing to consider here.

Looking at an apple falling a few meters in air friction is low - you abtain F=m a. That is ok for small velocities - at larger velocity friction is again the main thing (F=ma + rv) and then relativity (e.g. m seems to increase with v...), which is a more general description beyond Newton. Newton is still present inside relativity as a "small velocity simplification".

But there will be even more in future: Current work is about dark matter and such and in some years we will have a new formular. You should know that there is something very strange with F=m a. It is the "m". For some not yet clear reason that "inertial mass" m used in this formular is the same thing as the mass in gravitation today. I would not be to surprised if that would split in future e.g. for large distances in astronomy or such... but that depends solely on observations of nature and its precision.

Edit to get back to curve fitting: The precision of measurement is so important, because with smaller error bars it is more likely that you can reject the model you tested. As I mentioned above you can not proove a model but you can test by fitting, if the model is not valid. If the model is still valid in all observations in the world we may give it to the engineers to make things. But the effort is of course in physics to find new physics thus to disprove a model. To do so we need more precise measurements, smaller error bars and smaller covariance ellipses such that the data is 5 standard deviations away from model and we may savely say: Take that, model, you are gone for this!

[ChrisReynolds]

So what is the physical basis for F=MA?

There is no way to show a physical basis for this except to go collect some data and fit a curve to it.

From my understanding curve fitting was not used. Experiments were used, merely to demonstrate the proportionality of force and mass. Logic was then used to derive the equation. I'm no expert on Newton's Principia Mathematica, but I've never read of 'curve fitting' having any role in it.
http://en.wikipedia.org/wiki/Newton's_laws_of_motion#Newton.27s_2nd_Law

Principia Mathematica was published in 1687.

http://en.wikipedia.org/wiki/Founders_of_statistics
Playfair estabilished the underpinnnings of graphs in the late 1700s. Legendre and Gauss published the method of least squares in the early 1800s.

The relationship between observed variables had a role in the formulation of Ampere's Law and Faraday's Law. However those equations were reformulated by Maxwell as he involved some maths by Gauss and produced Maxwell's Equations, which describe the propagation of electromagnetic waves. That involved absolutely no curve fitting! It was a brilliant work of insight and mathematical induction/deduction - I get confused about those two - that's what comes of finding scientific philosophy boring.

Those equations underpin, antenna theory, propagation theory, transmission line (UHF and above) and waveguide theory. Without them being correct large parts of the technology we use would not exist.

[ChrisReynolds]

In the PIOMAS case as far as I understood that thing it is performed like in normal physics curve fitting: They can both test the quality of their choice of models with experimental data and also obtain some parameters with error bars - and you may use those parameters (e.g. regional thickness) for your nice engineering. But you know that they do not make any predictions about next years thickness. There is no physics in PIOMAS backing up any predictions about medium time scale future change of parameters like sea ice thickness as far as I know. So please be careful if you would like to try that.

Unless the changes to boundary conditions are so massive that they kick the system into a new stable state, stochastic variation (the most common of which is atmospheric weather) will always leave a gap between short term predictability (in the case of sea ice a window of weeks/months varying with the season), and long term prediction due to boundary conditions.

Is it possible therefore for a model to make accurate predictions? Yes, absolutely. In a qualitative sense, change the boundary conditions, e.g. increase (decrease) solar irradiance and it is guaranteed that the planet will warm (cool), likewise for CO2 increases. Both of which impact the basic energy balance of the planet. Putting numbers to it and you get into uncertainty, the best way then is to calculate a probability distribution.

The main reason PIOMAS isn't able to predict next years sea ice is the short term stochastic impact of weather. That applies to any attempt to predict the physical world into next year.

[Michael Hauber]

From my understanding curve fitting was not used. Experiments were used, merely to demonstrate the proportionality of force and mass. Logic was then used to derive the equation.

To demonstrate the proportionality of force and mass using an experiment you get some data, plot it, and fit a curve to it. 

[Michael Hauber]

The big difference between something that we think of as 'pure physics' such as F=MA, and the more questionable practices of curve fitting such as fitting various cyclic functions to modern temperature data and declaring a natural cycle is a question of accuracy of fit, and the possibility that the fit is a coincidence.

The wikipedia article on the speed of light tells us that in 1975 the accuracy of the speed of light was known to 4 parts in a billion.  Although measuring F=MA isn't exactly the same, it would be reasonable to assume that the best experiments we could do would have a roughly similar accuracy.  So if you plot this on a chart you wouldn't be able to see the gap between the line and the experimentally obtained data points unless you zoomed in a bunch.

And F=MA has been verified many many times.  Many engineering tasks, such as launching a rocket to hit a target depend on predictions based on this formula.  If the formula was wrong (enough to be measured), then every time this happens there is a chance that someone would notice the discrepancy between expectation and reality.  The curve has fit to a very high degree of accuracy a very great number of times.  It is ludicrously unlikely that this could happen only by chance.

In contrast with a typical questionable curve fitting exercise of temperature history, the fit is quite messy.  And if we are talking say a 60 year cycle, the fit is only over one or two repeats - which is something that can quite easily happen by coincidence.

A further case - how about curve fitting to determine that ENSO has a dramatic influence on global temperatures.  Does anyone seriously question this relationship?  Can anyone explain the physical basis of this relationship?  The fact that the relationship betweeen ENSO and global temperature has been repeated over maybe several dozen up and down cycles of ENSO gives reasonable confidence that the relationship is real and not just a 'curve fitting' exercise.

[ChrisReynolds]

From my understanding curve fitting was not used. Experiments were used, merely to demonstrate the proportionality of force and mass. Logic was then used to derive the equation.

To demonstrate the proportionality of force and mass using an experiment you get some data, plot it, and fit a curve to it.

But it doesn't look like it was the way it was originally worked out. Nor was it used by Maxwell in deriving his equations.

Back to sea ice. Consider the equations on page 8 of the following pdf.
http://www.unis.no/48_HSE/Info%20Lessons/Sea_Ice/Introduction%20to%20Sea%20Ice.pdf
No curves used to make them, just physical understanding and reason.

None of which changes the fact that there is no physical model behind the exponentials used to extrapolate zero sea ice in 2016 +/-3 years. However it is worth noting that the August zero crossing based on data to 2011 was 2018, now it is after 2020 with data to August 2014, see the first post in the Latest PIOMAS update thread.

[SATire]

From my understanding curve fitting was not used. Experiments were used, merely to demonstrate the proportionality of force and mass. Logic was then used to derive the equation. I'm no expert on Newton's Principia Mathematica, but I've never read of 'curve fitting' having any role in it.
http://en.wikipedia.org/wiki/Newton's_laws_of_motion#Newton.27s_2nd_Law

Principia Mathematica was published in 1687.

http://en.wikipedia.org/wiki/Founders_of_statistics
Playfair estabilished the underpinnnings of graphs in the late 1700s. Legendre and Gauss published the method of least squares in the early 1800s.

The relationship between observed variables had a role in the formulation of Ampere's Law and Faraday's Law. However those equations were reformulated by Maxwell as he involved some maths by Gauss and produced Maxwell's Equations, which describe the propagation of electromagnetic waves. That involved absolutely no curve fitting! It was a brilliant work of insight and mathematical induction/deduction - I get confused about those two - that's what comes of finding scientific philosophy boring.

Those equations underpin, antenna theory, propagation theory, transmission line (UHF and above) and waveguide theory. Without them being correct large parts of the technology we use would not exist.

Chris - please don't simulate beeing silly. Of course curve fitting is not the only thing done in physics. Especially in theoretical physics models are derived totally without any curve fitting - of course, since if you build something using the methode you need to test your construction that is flawed in the beginning...

So no curve fitting is used for constructing a new model. A new model is invented by imagination and by extensive use of mathematics in new ways. But 99% of such models imaginated are just bullshit - only 1 % of such new models will be named "theory" later and honored by e.g. a Nobel price.

Experimentalists use curve fitting to kick out the 99% of the models, which are not suitable to describe the nature - regardless if the mathematics behind it is so beautiful. The 1% of models which permanently can not be rejected by curve fitting to any data are the basis of our physics and our engineering.

So we have 2 simple facts:

1) Any model must survive the curve fitting to data from nature - so curve fitting is essential to test the physics model, if it is a desrciption of the nature or not.

2) No new model is derived by curve fitting because that is not possible. In any interval of data you may have an infinite number of functions possible to describe the data and those functions may vary dramatically outside that data intervall.

E.g. in sea-ice observations you have even problems to exclude the linear function from the possible functions describing the data. In this case all polynoms and nearly every other function may describe the data just as well. "A better fit" does not make much of a difference, since the probability that you must reject a specific function is not much different, e.g. less than 3 sigma and thus not worth to talk about. You can not proove anything by curve fitting if you do not have a model to test. So please stay with the curved stencil instead, if you have no model to test.

[SATire]

None of which changes the fact that there is no physical model behind the exponentials used to extrapolate zero sea ice in 2016 +/-3 years. However it is worth noting that the August zero crossing based on data to 2011 was 2018, now it is after 2020 with data to August 2014, see the first post in the Latest PIOMAS update thread.

Chris - you know it is the albedo-feedback model which is tested by fitting that exponential increase of sea ice loss to e.g. the PIOMAS data. We will see in a few years if we are able to reject that model. That would be an success, if for the first time we can exclude a model! (BTW. we do not exclude feedback in generall but we may exclude, that the "albedo feedback is the one and only driving force". So it will not be a very great surprise...)

[ChrisReynolds]

Yes, I know now you have explained the equation to me.

[Xyrus]

...

So no curve fitting is used for constructing a new model. A new model is invented by imagination and by extensive use of mathematics in new ways. But 99% of such models imaginated are just bullshit - only 1 % of such new models will be named "theory" later and honored by e.g. a Nobel price.

Experimentalists use curve fitting to kick out the 99% of the models, which are not suitable to describe the nature - regardless if the mathematics behind it is so beautiful. The 1% of models which permanently can not be rejected by curve fitting to any data are the basis of our physics and our engineering.

So we have 2 simple facts:

1) Any model must survive the curve fitting to data from nature - so curve fitting is essential to test the physics model, if it is a desrciption of the nature or not.

2) No new model is derived by curve fitting because that is not possible. In any interval of data you may have an infinite number of functions possible to describe the data and those functions may vary dramatically outside that data intervall.
 ...

Nonsense. A scientific model must do at least two things.

1. It must be plausible within the context of the phenomena being studied.

2. It must accurately and reliably reproduce and predict the behavior of the phenomena that is being studied under trivial and non-trivial circumstances.

By the time a scientist gets around to comparing their model to observational data, they've already eliminated "99%" of the nonsense simply by knowing their field and the research that has been done before them.

A plausible hypothesis is tested first against a trivial data set to verify that it can, at the very least, reproduce a very basic and/or simplified representation of the real phenomena (i.e, assume a spherical cow). A model that can't reproduce the phenomena when given known inputs is exceedingly unlikely to reproduce phenomena when fed a non-trivial scenario.

If the hypothesis survives the contrived scenario, it is tested against real data and observations. No model is perfect. No data set is perfect. Therefore, there will be deviations from the observations. Whether or not the model is useful depends on how well it behaves vs. the expected error as well as vs. other models (if they exist). In the case of climate models, for example, they're usually initialized with historical starting conditions and run forward to see how well they match the observational record.

Now we have a tested model that has shown skill. Can it be improved? Possibly. Model ensembles of perturbations can be run to see if parameters can be "tuned" to reduce model errors and better replicate observations. However, even this tuning is restricted (rain can't have negative mass, the earth can't move faster than the speed of light, etc.).

The "tuning" of a model is the only part that can be considered "curve fitting", and only in the loosest sense. The model is already established and validated. The tuning is simply trying to improve it, and you are restricted to what and how much you can tune.

Science isn't about matching curves. It's about why there are curves in the first place.

[cesium62]

We're all avid readers of Paul Krugman, right?  Anyway, he makes a point today, which, if you replace 'economist' with, perhaps, 'climate observer' seems relevant to this thread:

"
I guess the problem is that too many [climate observers] have the wrong attitude toward models. They’re not Truth; they’re intuition pumps, gadgets you use to clarify your story. You go badly wrong when you take them too seriously, and either forget that they’re just models or reject them because the world isn’t that simple.
"
http://krugman.blogs.nytimes.com/2014/09/11/their-own-imaginary-keynes-wonkish/?module=BlogPost-ReadMore&version=Blog%20Main&action=Click&contentCollection=Opinion&pgtype=Blogs&region=Body#more-37498.

Now if I can just figure out if he is scolding me or reinforcing a point I was trying to make...